Core-collapse supernovae have challenged theorists and computational
science for half a century. Such explosions are the source of many of the
heavy elements in the Universe and the birthplace of neutron stars and
stellar-mass black holes. However, determining the mechanism of explosion
remains the key goal of theory. Though the synergistic operation of
turbulence and neutrino heating seems implicated, and multi-dimensional
simulations with some physical fidelity that have provided insight, we
have yet to reproduce the phenomenon theoretically. In this talk, I will
review the goals of supernova theory, the state of the field, and the
contending explosion models. In the process, I will highlight the
computational astrophysics that has been applied to date, and that may be
necessary in the future to credibly unravel this mystery.